1. Technical Field
In general, the present disclosure relates to nuclear medical imaging. More particularly, the disclosure relates to Positron Emission Tomography (PET) imaging and accurate estimation of expected random coincidences through variance reduction (VR).
2. General Background of the Invention
Nuclear medicine is a unique specialty wherein radiation emission is used to acquire images that show the function and physiology of organs, bones or tissues of the body. The technique of acquiring nuclear medicine images entails first introducing radiopharmaceuticals into the body—either by injection or ingestion. These radiopharmaceuticals are attracted to specific organs, bones, or tissues of interest. Upon arriving at their specified area of interest, the radiopharmaceuticals produce gamma photon emissions, which emanate from the body and are then captured by a scintillation crystal. The interaction of the gamma photons with the scintillation crystal produces flashes of light, which are referred to as “events.” Events are detected by an array of photo detectors (such as photomultiplier tubes), and their spatial locations or positions are then calculated and stored. In this way, an image of the organ or tissue under study is created from detection of the distribution of the radioisotopes in the body.
One particular nuclear medicine imaging technique is known as positron emission tomography, or PET. PET is used to produce images for diagnosing the biochemistry or physiology of a specific organ, tumor or other metabolically active site. The measurement of tissue concentration using a positron emitting radionuclide is based on coincidence detection of the two gamma photons arising from a positron annihilation. When a positron is annihilated by an electron, two 511 keV gamma photons are simultaneously produced and travel in approximately opposite directions. Gamma photons produced by an annihilation event can be detected by a pair of oppositely disposed radiation detectors capable of producing a signal in response to the interaction of the gamma photons with a scintillation crystal. Annihilation events are typically identified by a time coincidence between the detection of the two 511 keV gamma photons in the two oppositely disposed detectors; i.e., the gamma photon emissions are detected virtually simultaneously by each detector. When two oppositely disposed gamma photons each strike an oppositely disposed detector to produce a time coincidence event, they also identify a line-of-response (LOR) along which the annihilation event has occurred. An example of a PET method and apparatus is described in U.S. Pat. No. 6,858,847, which patent is incorporated herein by reference in its entirety.
FIG. 1 is a graphic representation of a line of response. An annihilation event 140 occurring in imaged object mass 130 may emit two simultaneous gamma rays (not shown) traveling substantially 180° apart. The gamma rays may travel out of scanned mass 130 and may be detected by block detectors 110A and 110B, where the detection area of the block detector defines the minimum area or maximum resolution within which the position of an incident gamma ray may be determined. Since block detectors 110A and 110B are unable to determine precisely where the gamma rays were detected within this finite area, the LOR 120 connecting block detectors 110A and 10B may actually be a tube with its radius equal to the radius of block detectors 110A and 110B. Similar spatial resolution constraints are applicable to other types of detectors, such as photomultiplier tubes.
To minimize data storage requirements, clinical projection data are axially compressed, or mashed, to within a predetermined span. With a cylindrical scanner, which has translational symmetry, the geometrical blurring resulting from axial compression may be modeled by projecting a blurred image into LOR space, followed by axial compression. This eliminates the storage of the axial components, and special algorithms have been developed to incorporate system response. The system response modeling then will allow the use of standard reconstruction algorithms such as Joseph's Method, and a reduction of data storage requirements.
The LOR defined by the coincidence events are used to reconstruct a three-dimensional distribution of the positron-emitting radionuclide within the patient. In two-dimensional PET, each 2D transverse section or “slice” of the radionuclide distribution is reconstructed independently of adjacent sections. In fully three-dimensional PET, the data are sorted into sets of LOR, where each set is parallel to a particular detector angle, and therefore represents a two dimensional parallel projection p(s, Φ) of the three dimensional radionuclide distribution within the patient, where “s” corresponds to the displacement of the imaging plane perpendicular to the scanner axis from the center of the gantry, and “Φ” corresponds to the angle of the detector plane with respect to the x axis in (x, y) coordinate space (in other words, Φ corresponds to a particular LOR direction).
Coincidence events are integrated or collected for each LOR and stored in a sonogram. In this format, a single fixed point in f(x, y) traces a sinusoid in the sonogram. In each sonogram, there is one row containing the LOR for a particular azimuthal angle Φ; each such row corresponds to a one-dimensional parallel projection of the tracer distribution at a different coordinate along the scanner axis. This is shown conceptually in FIG. 2.
FIG. 3 depicts mashing and axial rebinning of LOR into sonograms. Five of six LORs are combined in one axial span complex. Four LORs are combined together transaxially.
The ordinary Poisson iterative reconstruction algorithm is now widely-used in the PET imaging field to produce clinically useful data from the sonograms. The model behind the algorithm considers prompt coincidences as independent Poisson variables, with expected values equal to the forward projection of the reconstructed image added to the expected scattered and random contributions. The expected random contribution of line-of-response (LOR) with indexes ij, (where i and j denote crystals in coincidence), can be estimated from a crystal singles rate, according to the classical random rate equation: rij=2τ si sj,  (1)where si is the mean single rate for a given crystal i, and 2τ is the coincidence time window. This approach requires the acquisition of singles rates at the crystal level. A potential problem with this method is that real random coincidences may have additional dead time when compared to singles (due e.g. to multiplexing).
Alternatively, random projection data can be acquired separately, since random events are part of the list mode data. In the latest case, random data are noisy and mean values for the random need to be estimated. We redefine the singles rate as si=√{square root over (2τ si)}, so one does not need to keep track of the multiplicative factor. The problem is the estimation of these single rates from noisy random data.
There are a number of methods that may be used to estimate singles from uncompressed LOR or list mode delayed coincidence data. Analytical methods use equation (1), which is averaged over selected groups of crystals in full coincidence. A measured, mashed group of random data, serves as a good approximation of the mean value. Then the problem is reduced to finding values for the smaller set singles over group of crystals. The Casey smoothing method does not estimate singles rates, but rather estimates expected random data based on large groups of random coincidences. The method from Defrise estimates group singles rates from a set of nonlinear equations using only part of the available data. Both methods are exact and useful for list mode data processing, where large groups of crystals can be defined by the user. The fan sum method is not exact and leads to bias with non-uniformly distributed activity. An analytical method allows for fast pre-processing of data. This is essential in the clinical environment, where image reconstruction consumes most of the available processing time and VR must be performed for each patient bed acquisition. However, the choice of the crystal group in full coincidence can be a complicated task. For example, a group of crystals is commonly chosen to be a detector head. However, on most clinical scanners, only data up to a maximum ring difference are passed to list mode, since the very oblique LORs are not used in reconstruction. Therefore, the detector heads are not in full coincidence and the choice of crystal groups becomes cumbersome. In addition to this, the group size influences the final result.
On the other hand, crystal singles can be estimated by the Maximum Likelihood (ML) approach, where random coincidences are Poisson variables with expected values provided by equation (1). The iterative approach has the advantage of versatility, where all available data are easily accommodated. It is truly a statistical method. This approach results in solving a large system of nonlinear equations iteratively, and is consequently potentially resource consuming. By examining the structure of the objective function in case of list mode data acquisition it was shown that random data have sufficient statistics. In order to estimate singles rate only fan sum of random data are needed. This interesting property has the potential to save histogramming bandwidth when the fan sums are computed at coincidence processor level. The coordinate ascent and surrogate functions monotonic algorithm were developed. Such algorithms are quite fast since there is no need to model mean random data in iterative process, only their fan sum.
In a clinical environment, PET data are acquired with some type of compression to reduce data size. While it is important for prompt data, delayed coincidences are processed by hardware rebinners in a similar way. Two types of data compression are considered in FIG. 3. The first compression is axial (span) compression, where the number of oblique planes is significantly reduced by combining the LOR of similar polar angles. The second compression is transaxial mashing, where the number of sonogram views is reduced by combining the LOR of similar azimuthal angles. Data compression does change the basic random equation (1) and causes the mentioned methodologies to no longer be applicable.
The goal behind of this paper was development of simple update equation iterative algorithms to estimate singles from compressed random data. The first algorithm is Coordinate Descent (CD), which possesses an advantage of fast convergence. Unfortunately, a closed form updated equation can be designed only for a Least Squares objective function. A simultaneous update (SU) algorithm is derived by a surrogate function construction. We consider two SU algorithms. One optimizes the same objective function as a CD algorithm and another optimizes the Poisson Likelihood function.
Unfortunately, in case of compressed data, iterative algorithms need to compare measured random data against modeled random data. Construction of mean random data takes significant time, so algorithms are not computationally fast, comparing with list mode data acquisition, where only fan sum are needed to be constructed.